A number which is simultaneously a pentagonal number 
and triangular number 
. Such numbers exist when
 |
(1)
|
Completing the square gives
 |
(2)
|
Substituting 
and 
gives the Pell-like quadratic Diophantine equation
 |
(3)
|
which has solutions 
, (19, 11), (71, 41), (265, 153), .... In terms of
,
these give (1, 1), (10/3,5), (12, 20), (133/3, 76), (165, 285), ..., of which the
whole number solutions are 
, (12, 20), (165, 285), (2296, 3976), ... (OEIS A046174 and A046175),
corresponding to the pentagonal triangular numbers 1, 210, 40755, 7906276, 1533776805,
... (OEIS A014979).
